Simpliﬁed Interception/Evaporation Model

: It is known that at the event scale, evaporation losses of rainfall intercepted by canopy are a few millimeters, which is often not much in comparison to other stocks in the water balance. Nevertheless, at yearly scale, the number of times that the canopy is ﬁlled by rainfall and then depleted can be so large that the interception ﬂux may become an important fraction of rainfall. Many accurate interception models and models that describe evaporation by wet canopy have been proposed. However, they often require parameters that are difﬁcult to obtain, especially for large-scale applications. In this paper, a simpliﬁed interception/evaporation model is proposed, which considers a modiﬁed Merrian model to compute interception during wet spells, and a simple power-law equation to model evaporation by wet canopy during dry spells. Thus, the model can be applied for continuous simulation, according to the sub hourly rainfall data that is appropriate to study both processes. It is shown that the Merrian model can be derived according to a simple linear storage model, also accounting for the antecedent intercepted stored volume, which is useful to consider for the suggested simpliﬁed approach. For faba bean cover crop, an application of the suggested procedure, providing reasonable results, is performed and discussed.


Introduction
According to Brutsaert [1], the interception process is determined by the rainfall fraction that moistens vegetation and that is temporarily stored on it, before evaporating. When the vegetation cover is fully saturated, the interception storage capacity is achieved. In practice, the interception storage capacity is denoted as rainfall left on the canopy at the end of the rainfall after all drip has ceased [2]. The water stored on the canopy may evaporate soon after, thus short-circuiting the hydrologic cycle.
Although most surfaces can store only a few millimeters of rainfall, which is often not much in comparison to other stocks in the water balance, interception is generally a significant process and its impact becomes evident at longer time scales [3]. Thus, interception storage is generally small, but the number of times that the storage is filled and depleted can be so large that the evaporation losses by wet canopy may become of the same order of magnitude as the transpiration flux [4].
Evaporation flux by canopy exerts a negative effect on plant water consumption by preventing water from reaching the soil surface, thus the plant roots [2,3]. In contrast, the remaining rainfall (i.e., the net rainfall) reaches the soil surface either as throughfall or by flowing down branches and stems as stemflow. Throughfall is the fraction of water that reaches the soil surface directly through the canopy gaps without hitting the canopy surfaces, or indirectly through dripping from the leaves and branches [5].
The interception may also exert important effects on surface runoff [6], providing a certain delay compared with the time of the beginning of the rain. For the Dunnian mechanism of runoff generation, Baiamonte [7] showed the effect induced by the interception process on the delay time, and emphasized that the effect is more frequent for well-drained soils [8,9] in humid regions, for low rainfall intensities and high groundwater table, when infiltration capacity exceeds the rainfall intensity. Indeed, the latter conditions also occurs combines a modified exponential equation applied and tested by Merrian [18], accounting for the antecedent volume stored on the canopy, and a simple power-law equation to compute evaporation by wet canopy. We are aware that the considered approach is far from the sophisticated physically based developments that were performed to quantify interception and evaporations losses. However, the latter may require many parameters that are not easy to determine. It is shown that the simplified parsimonious approach may lead to a reasonable quantification of this important component of the hydrologic cycle, which can be useful when a rough estimate is required, in absence of a detailed characterization of the canopy and of the climate conditions. It is also shown that the Merrian model can be derived by considering a simple linear storage model. For faba bean cover crop, an application of the suggested procedure is performed and discussed.

Rainfall Data Set, Wet and Dry Spells
Rainfall time series were analyzed according to Agnese et al. [32], who applied the discrete three-parameter Lerch distribution to model the frequency distribution of interarrival times, IT, derived from daily precipitation time-series, for the Sicily region, in Italy. Agnese et al. [32] showed a good fitting of the Lerch distribution, thus evidencing the wide applicability of this kind of distribution [33], also allowing us to jointly model dry spells, DS, and wet spells, WS.
Since this work aimed at modelling by continuous rainfall data series the interception losses, during the WS and the evaporation losses during DS, only the frequency distribution of the two latter were considered, which are defined in the following, according to Agnese et al. [32].
For any rainfall data series, the ten minutes temporal scale, τ, which is appropriate to model both interception and evaporation processes, could be considered in order to describe clustering and intermittency characters of continuous rainfall data series.
Let H = {h 1 , h 2 , . . . , h n }, a time-series of rainfall data of size n, spaced at uniform time-scale τ. The sub-series of H can be defined as the event series, E {t 1 , t 2 , . . . , t nE }, where n E (0 < n E < n) is the size of E, which is an integer multiple of time-scale τ. The succession constituted by the times elapsed between each element of the E series, with exception of the first one and the immediately preceding one, is defined as the inter-arrival time-series, IT {T 1 , T 2 , . . . , T nE }, with size n E−1 (Figure 1). It can be observed the sequence of dry spells, DS, can be derived from IT dataset by using the relationship {DS k } = {T k } − 1 for any T k > 1. In this paper, we explored the rainfall partitioning in net rainfall and evaporation losses by canopy, by using a very simplified sketch of the interception process, which combines a modified exponential equation applied and tested by Merrian [18], accounting for the antecedent volume stored on the canopy, and a simple power-law equation to compute evaporation by wet canopy. We are aware that the considered approach is far from the sophisticated physically based developments that were performed to quantify interception and evaporations losses. However, the latter may require many parameters that are not easy to determine. It is shown that the simplified parsimonious approach may lead to a reasonable quantification of this important component of the hydrologic cycle, which can be useful when a rough estimate is required, in absence of a detailed characterization of the canopy and of the climate conditions. It is also shown that the Merrian model can be derived by considering a simple linear storage model. For faba bean cover crop, an application of the suggested procedure is performed and discussed.

Rainfall Data Set, Wet and Dry Spells
Rainfall time series were analyzed according to Agnese et al. [32], who applied the discrete three-parameter Lerch distribution to model the frequency distribution of interarrival times, IT, derived from daily precipitation time-series, for the Sicily region, in Italy. Agnese et al. [32] showed a good fitting of the Lerch distribution, thus evidencing the wide applicability of this kind of distribution [33], also allowing us to jointly model dry spells, DS, and wet spells, WS.
Since this work aimed at modelling by continuous rainfall data series the interception losses, during the WS and the evaporation losses during DS, only the frequency distribution of the two latter were considered, which are defined in the following, according to Agnese et al. [32].
For any rainfall data series, the ten minutes temporal scale, τ, which is appropriate to model both interception and evaporation processes, could be considered in order to describe clustering and intermittency characters of continuous rainfall data series.
Let H = {h1, h2, …, hn}, a time-series of rainfall data of size n, spaced at uniform timescale τ. The sub-series of H can be defined as the event series, E {t1, t2, …, tnE}, where nE (0 < nE < n) is the size of E, which is an integer multiple of time-scale τ. The succession constituted by the times elapsed between each element of the E series, with exception of the first one and the immediately preceding one, is defined as the inter-arrival time-series, IT {T1, T2, …, TnE}, with size nE−1 ( Figure 1). It can be observed the sequence of dry spells, DS, can be derived from IT dataset by using the relationship {DSk} = {Tk} − 1 for any Tk > 1.  Figure 1 shows an example of a sequence of wet and dry spells. In the context of this work, which aims at modelling the interception process during WS and the evaporation process from the canopy, as previously observed, only the frequency distribution of wet spells (WS) and dry spells (DS) were derived.
For the 2009 rainfall data series of Fontanasalsa station (Trapani, 37°56′37″ N, 12°33′12″ E, western Sicily, Italy), which will be considered for an example application, Figure 2 shows the complete characteristics of the rainfall regime. In particular, Figure 2a  Figure 1 shows an example of a sequence of wet and dry spells. In the context of this work, which aims at modelling the interception process during WS and the evaporation process from the canopy, as previously observed, only the frequency distribution of wet spells (WS) and dry spells (DS) were derived.
For the 2009 rainfall data series of Fontanasalsa station (Trapani, 37 • 56 37 N, 12 • 33 12 E, western Sicily, Italy), which will be considered for an example application, Figure 2 shows the complete characteristics of the rainfall regime. In particular, Figure 2a describes the DS distribution of frequency, F, whereas Figure 2b plots the WS distribution associated with the cumulated rainfall depth collected with 10 min time resolution. Both figures also illustrate the frequency of non-exceedance, corresponding the selected time resolution (τ = 1 = 10 min) that equals 0.189 for DS and much higher for WS (0.596), indicating that a high fraction of rainfall with WS = 1 occurs, which could play an important role in the interception process. Figure 2b also plots the rainfall depth distribution corresponding to WS, where the frequency is calculated with respect to the yearly rainfall depth, h year , which equals 885.2 mm. Therefore, in Figure 2b, to WS = 1 (F 1 = 0.115) corresponds a rainfall depth equals to 101.6 mm that, if associated with subsequent large enough DS, may potentially evaporate from the canopy. describes the DS distribution of frequency, F, whereas Figure 2b plots the WS distribution associated with the cumulated rainfall depth collected with 10 min time resolution. Both figures also illustrate the frequency of non-exceedance, corresponding the selected time resolution (τ = 1 = 10 min) that equals 0.189 for DS and much higher for WS (0.596), indicating that a high fraction of rainfall with WS = 1 occurs, which could play an important role in the interception process. Figure 2b also plots the rainfall depth distribution corresponding to WS, where the frequency is calculated with respect to the yearly rainfall depth, hyear, which equals 885.2 mm. Therefore, in Figure 2b, to WS = 1 (F1 = 0.115) corresponds a rainfall depth equals to 101.6 mm that, if associated with subsequent large enough DS, may potentially evaporate from the canopy.

Interception Rate and Stored Volumes during Wet Spells
The interception Merrian [18]'s model states that: where S (mm) is the interception capacity of the canopy, and ICS (mm) is the interception storage volume corresponding to the cumulated rainfall volume R (mm). Equation (1) can be applied for dry initial condition, i.e., when the no water volume is stored on the canopy at the beginning of the rainfall. However, it is demonstrated here that the Merrian model can be derived by considering a simple linear stored model, also used in other hydrological contexts [34], which made it possible to also account for the antecedent interception volume. The latter is useful for the applications, when in between two consecutive WSs, DS is not long enough to result in full evaporation of the rainfall volume intercepted by the canopy starting from the end of a WS. Thus, a residual water volume, ICS0, is still stored on the canopy at the beginning of the subsequent WS, and it needs to be considered as initial condition. Therefore, for the purpose of this study, which aims at estimating interception losses during WS and DS sequences of different amount and duration, Equation (1) needs to be extended to different antecedent water interception storage volume, ICS0 (mm), before a new WS takes place, after a DS when evaporation process ceases.
Consider a simple linear reservoir, miming the interception storage volume ( Figure  3). A stationary rainfall of intensity r (mm h −1 ) uniformly distributed over the canopy, is applied to the reservoir. In the time dt, the interception storage, ICS (mm), stored in the canopy equals dICS. The interception capacity, i.e., the maximum rainfall volume that can be stored on the canopy, is denoted as S (mm). The duration of rainfall that needs to

Interception Rate and Stored Volumes during Wet Spells
The interception Merrian [18]'s model states that: where S (mm) is the interception capacity of the canopy, and ICS (mm) is the interception storage volume corresponding to the cumulated rainfall volume R (mm). Equation (1) can be applied for dry initial condition, i.e., when the no water volume is stored on the canopy at the beginning of the rainfall. However, it is demonstrated here that the Merrian model can be derived by considering a simple linear stored model, also used in other hydrological contexts [34], which made it possible to also account for the antecedent interception volume. The latter is useful for the applications, when in between two consecutive WSs, DS is not long enough to result in full evaporation of the rainfall volume intercepted by the canopy starting from the end of a WS. Thus, a residual water volume, ICS 0 , is still stored on the canopy at the beginning of the subsequent WS, and it needs to be considered as initial condition. Therefore, for the purpose of this study, which aims at estimating interception losses during WS and DS sequences of different amount and duration, Equation (1) needs to be extended to different antecedent water interception storage volume, ICS 0 (mm), before a new WS takes place, after a DS when evaporation process ceases.
Consider a simple linear reservoir, miming the interception storage volume ( Figure 3). A stationary rainfall of intensity r (mm h −1 ) uniformly distributed over the canopy, is applied to the reservoir. In the time dt, the interception storage, ICS (mm), stored in the canopy equals dICS. The interception capacity, i.e., the maximum rainfall volume that can be stored on the canopy, is denoted as S (mm). The duration of rainfall that needs to achieve the saturation of the canopy, without dripping out from the reservoir, t s (h), can be expressed as: From a physical point of view, ts is not very meaningful since it describes the duration of rainfall that is required for the canopy saturation, for dry antecedent conditions (ICS0 = 0), and with no losses from the reservoir, e.g., no dripping and no net rainfall intensity pn (mm h −1 ), which actually occurs ( Figure 3). However, ts represents a characteristic time of the interception process that it is useful to introduce for the following derivations. By assuming that the water volume stored in the reservoir linearly varies with the output net rainfall intensity, pn, according to the time ts (Equation (2)), the balance of the water volumes can be expressed as: Separation of variables may be used to solve the ordinary differential equation: By assuming as initial conditions: Integration of Equation (4) provides: (6) Solving Equation (6) yields: + Equation (7) can be made explicit to derive the net rainfall intensity, pn: Of course, knowledge of Equation (8) makes it possible to determine the interception intensity, ics (mm/h): From a physical point of view, t s is not very meaningful since it describes the duration of rainfall that is required for the canopy saturation, for dry antecedent conditions (ICS 0 = 0), and with no losses from the reservoir, e.g., no dripping and no net rainfall intensity p n (mm h −1 ), which actually occurs ( Figure 3). However, t s represents a characteristic time of the interception process that it is useful to introduce for the following derivations.
By assuming that the water volume stored in the reservoir linearly varies with the output net rainfall intensity, p n , according to the time t s (Equation (2)), the balance of the water volumes can be expressed as: Separation of variables may be used to solve the ordinary differential equation: By assuming as initial conditions: Integration of Equation (4) provides: Solving Equation (6) yields: Equation (7) can be made explicit to derive the net rainfall intensity, p n : Of course, knowledge of Equation (8) makes it possible to determine the interception intensity, ics (mm/h): Due to the assumption of a linear storage model, the antecedent net rainfall intensity, p n0 , in Equations (8) and (9), is linked to the antecedent interception volume, ICS 0 , by: For a compacted graphical illustration, a constant rainfall intensity, r, and a zero time antecedent condition t 0 = 0 can be assumed. Thus, Equations (8) and (9) normalized with respect to r can be expressed as: where in Equations (8) and (9) t s was substituted by Equation (2), and r by R t, with R (mm) the cumulated rainfall depth. Equations (11) and (12) are graphed in Figure 4a,b, respectively. For any fixed S, and for a fixed antecedent interception storage volume, ICS 0 , Figure 4a shows that, at increasing R, the net rainfall intensity, p n , attains the gross rainfall intensity r (p n /r~1), and that, at increasing ICS 0 , the r achievement is slower and slower, as it could be expected. Of course, the normalized interception intensity ics/r plotted in Figure 4b provides complementary curves. Due to the assumption of a linear storage model, the antecedent net rainfall intensity, pn0, in Equations (8) and (9), is linked to the antecedent interception volume, ICS0, by: For a compacted graphical illustration, a constant rainfall intensity, r, and a zero time antecedent condition t0 = 0 can be assumed. Thus, Equations (8) and (9) normalized with respect to r can be expressed as: where in Equations (8) and (9) ts was substituted by Equation (2), and r by R t, with R (mm) the cumulated rainfall depth. Equations (11) and (12) are graphed in Figure 4a and 4b, respectively. For any fixed S, and for a fixed antecedent interception storage volume, ICS0, Figure 4a shows that, at increasing R, the net rainfall intensity, pn, attains the gross rainfall intensity r (pn/r ~ 1), and that, at increasing ICS0, the r achievement is slower and slower, as it could be expected. Of course, the normalized interception intensity ics/r plotted in Figure 4b provides complementary curves. In order to determine the interception volume, ICS (mm), as a function of its antecedent amount, ICS0 (mm), because of the assumption of a linear storage model (Equation (10)), it can be easily observed that Equation (11) also matches the interception storage volume, ICS, normalized with respect to the infiltration capacity, S: as displayed in the vertical axis label of Figure 4a. Therefore, Equation (13) is formally equivalent to Equation (11), but with a different meaning of the output parameters.
It is noteworthy to observe that, for dry antecedent condition (ICS0 = 0), Equation (13) matches the interception storage model (Equation (1)) proposed by Merrian [18], which was developed only for dry initial conditions, as previously observed.
Equation (13) could also be derived analytically, with a lesser extend from a physical point of view, than that provided by the simple linear stored model previously described. Indeed, denoting R0 the antecedent rainfall volume corresponding to ICS0, the Merrian's model (Equation (1)) yields: In order to determine the interception volume, ICS (mm), as a function of its antecedent amount, ICS 0 (mm), because of the assumption of a linear storage model (Equation (10)), it can be easily observed that Equation (11) also matches the interception storage volume, ICS, normalized with respect to the infiltration capacity, S: as displayed in the vertical axis label of Figure 4a. Therefore, Equation (13) is formally equivalent to Equation (11), but with a different meaning of the output parameters.
It is noteworthy to observe that, for dry antecedent condition (ICS 0 = 0), Equation (13) matches the interception storage model (Equation (1)) proposed by Merrian [18], which was developed only for dry initial conditions, as previously observed.
Equation (13) could also be derived analytically, with a lesser extend from a physical point of view, than that provided by the simple linear stored model previously described.
Indeed, denoting R 0 the antecedent rainfall volume corresponding to ICS 0 , the Merrian's model (Equation (1)) yields: The antecedent rainfall volume, R 0 , can be expressed as a function of ICS 0 by using Equation (1): Substituting Equation (15) into Equation (14), and normalizing ICS with respect to S, provides Equation (13), which was to be demonstrated.
As an example, for S = 0.5 mm and ICS 0 = 0, and for a constant rainfall intensity r = 1.75 mm/h, Figure 5 graphs ICS vs. the cumulated rainfall R (solid black line). For R 0 = 0.7 mm, corresponding to ICS 0 = 0.377 mm, Equation (14) is plotted in Figure 5 (red line), which of course is not physical meaningful for R < 0 (dashed line).
The antecedent rainfall volume, R0, can be expressed as a function of ICS0 by using Equation (1): Substituting Equation (15) into Equation (14), and normalizing ICS with respect to S, provides Equation (13), which was to be demonstrated.
As an example, for S = 0.5 mm and ICS0 = 0, and for a constant rainfall intensity r = 1.75 mm/h, Figure 5 graphs ICS vs. the cumulated rainfall R (solid black line). For R0 = 0.7 mm, corresponding to ICS0 = 0.377 mm, Equation (14) is plotted in Figure 5 (red line), which of course is not physical meaningful for R < 0 (dashed line). For ICS0 = 0.377 mm, Equation (13) is also plotted in Figure 5 (round circles) showing that Equation (13) matches Equation (14). Equation (13) will be used in the following to calculate the interception storage volume starting from any ICS0 value.

Evaporation by Wet Canopy during Dry Spells
During a dry spell, DS, starting from an antecedent storage volume, ICS0 > 0, the evaporation process from the wet canopy takes place. When the canopy is wetted by rain, evaporation of intercepted rainfall is largely a physical process that does not depend on the functioning of stomata. According to a realistic description of evaporation from canopies the Penman-Monteith equation [23,35] could be used, by imposing zero the surface (or canopy) resistance.
However, a different approach is used here, which is based on the physical circumstance highlighted well by Babu et al. [36] that evaporation by wet canopy comprises two stages. First, the drying process involves removal of unbound (free) moisture from the surface, and second it involves removal bound moisture from the interior of the leaf till a defined limit, corresponding to a critical moisture content. Apart from the second stage, which refers to water consumption by the canopy, and thus it is beyond the purpose of this study, the first stage comprises (i) a "preheat period", where the drying speed quickly For ICS 0 = 0.377 mm, Equation (13) is also plotted in Figure 5 (round circles) showing that Equation (13) matches Equation (14). Equation (13) will be used in the following to calculate the interception storage volume starting from any ICS 0 value.

Evaporation by Wet Canopy during Dry Spells
During a dry spell, DS, starting from an antecedent storage volume, ICS 0 > 0, the evaporation process from the wet canopy takes place. When the canopy is wetted by rain, evaporation of intercepted rainfall is largely a physical process that does not depend on the functioning of stomata. According to a realistic description of evaporation from canopies the Penman-Monteith equation [23,35] could be used, by imposing zero the surface (or canopy) resistance.
However, a different approach is used here, which is based on the physical circumstance highlighted well by Babu et al. [36] that evaporation by wet canopy comprises two stages. First, the drying process involves removal of unbound (free) moisture from the surface, and second it involves removal bound moisture from the interior of the leaf till a defined limit, corresponding to a critical moisture content. Apart from the second stage, which refers to water consumption by the canopy, and thus it is beyond the purpose of this study, the first stage comprises (i) a "preheat period", where the drying speed quickly increases, and then (ii) a "constant rate period", where evaporation takes place at the outside surface for the removal of unbound moisture (free water) from the surface of the leaf [36].
The evaporation mechanism by wet canopy of the first stage could be also described by the physical equations requiring the knowledge of climatic parameters and structure parameters of the canopy. However, in agreement with the simple sketch also considered for the interception process, in this simplified study the first stage evaporation mechanism is described by a simple power-law, according to two parameters. A similar power-law equation was also considered by Black et al. [37] to model the cumulative evaporation of an initially wet, deep soil. For the same power-law equation, Ritchie [38] reports the experimental parameters obtained by other researchers for different soils.
One limited experimental campaign, described in the next section for the faba bean, supported this choice and revealed that for fixed outdoor air temperature, T ex ( • C) the cumulated evaporation volume, per unit leaf surface area, E (mm), could be actually described by the power-law equation: where t (h) is the time spent after the canopy interception capacity, S, is achieved, m is a scale parameter and n a shape parameter to be determined by experimental measurements. In order to upscale Equation (16) to any values of air temperature, a regional equation developed for the Sicily region was considered [39]: where E m (mm) is the monthly evaporation depth and T m ( • C) is the monthly temperature. By scaling Equation (16) where the leaf area index, LAI, was introduced, accounting for the actual leaf surface from which the water evaporates. Of course, Equation (18) gives the same experimental evaporation amount derived by Equation (17), when the experimental air temperature, T ex , is equal to T m . It should be noted that Equation (18) does not account for wind speed, thus evaporation losses are linked to the wind experimental conditions for which m and n parameters are determined, otherwise losses are underestimated or overestimated for wind speeds lower and higher than the experimental ones, respectively. Moreover, application of this procedure in regions different from Sicily would require modifying Equations (17) and (18).
For LAI = 2 and T ex = 18 • C, and for fixed experimental values of m and n parameters, Figure 6 shows evaporation losses, E, during the time, with the air temperature, T m , as a parameter.
In order to apply the suggested procedure, according to the discrete nature of rainfall, the cumulated evaporation volume (Equation (18)) needs to be expressed in discrete terms by accounting, as per the interception model, for the antecedent conditions, at the aim to determine the E fraction, ∆E, which occurs during the dry spells, DS, starting from t 0 : where the antecedent initial condition, t 0 , refers to the end of the wet spell, WS, when the evaporation process starts. Thus, t 0 needs to be calculated by Equation (19), by assuming that the evaporation initial condition, E 0 , equals the interception capacity minus the water stored in the canopy as interception, S − ICS: In order to apply the suggested procedure, according to the discrete nature of rainfall, the cumulated evaporation volume (Equation (18)) needs to be expressed in discrete terms by accounting, as per the interception model, for the antecedent conditions, at the aim to determine the E fraction, ∆E, which occurs during the dry spells, DS, starting from t0: ,.-.

+
where the antecedent initial condition, t0, refers to the end of the wet spell, WS, when the evaporation process starts. Thus, t0 needs to be calculated by Equation (19), by assuming that the evaporation initial condition, E0, equals the interception capacity minus the water stored in the canopy as interception, S − ICS: By using Equations (18) and (20), an example of ∆E calculation (Equation (19)), which is useful for the applications that will be shown, is performed here. Let assume m = 0.047, n = 0.657, Tex = 22 °C (Figure 6), and an evaporation loss ∆E for a mean temperature of Tm = 22 °C needs to be determined. The interception capacity S equals 1.5 mm, and the interception volume stored in the canopy ICS when the evaporation process takes place equals 0.8 mm, thus the difference S − ICS = 0.7 mm mimics the amount water loss due to a virtual evaporation till E0 ( Figure 6). The corresponding initial condition t0, calculated by Equation (20) provides 12 h. The pair (t0, E0) is illustrated in Figure 6 (point A).
Assuming that ∆E needs to be calculated after a DS = 3 h, Equation (19) yields ∆E = 0.111 mm, which corresponds to the segment B-C in Figure 6, also indicating an evaporation loss E = E0 + ∆E = 0.816 mm (point C).
In order to consider that ∆E computation is limited by the antecedent volume stored water on the canopy, ICS, actually available to evaporate, the following condition was imposed, and a corrected ∆E, denoted as Δ$ * evaluated: By using Equations (18) and (20), an example of ∆E calculation (Equation (19)), which is useful for the applications that will be shown, is performed here. Let assume m = 0.047, n = 0.657, T ex = 22 • C (Figure 6), and an evaporation loss ∆E for a mean temperature of T m = 22 • C needs to be determined. The interception capacity S equals 1.5 mm, and the interception volume stored in the canopy ICS when the evaporation process takes place equals 0.8 mm, thus the difference S − ICS = 0.7 mm mimics the amount water loss due to a virtual evaporation till E 0 ( Figure 6). The corresponding initial condition t 0 , calculated by Equation (20) provides 12 h. The pair (t 0 , E 0 ) is illustrated in Figure 6 (point A).
Assuming that ∆E needs to be calculated after a DS = 3 h, Equation (19) yields ∆E = 0.111 mm, which corresponds to the segment B-C in Figure 6, also indicating an evaporation loss E = E 0 + ∆E = 0.816 mm (point C).
In order to consider that ∆E computation is limited by the antecedent volume stored water on the canopy, ICS, actually available to evaporate, the following condition was imposed, and a corrected ∆E, denoted as ∆E * evaluated: In both interception and evaporation models previously introduced, neither the canopy actually covering the field (cover fraction), nor the proportion of rain which falls through the canopy without striking a surface (throughfall), were taken into account. In the next section, the latter issues are considered and then the water mass balance is analyzed.

Water Mass Balance
The fraction of ground covered by the canopy [40] plays a fundamental role in estimating interception losses, as well as the proportion of rain that falls through the canopy without striking any surface. In the following, in order to analyze the water mass balance, C F denotes the fraction of ground covered by the canopy and p is the free throughfall coefficient [19,20]. Assuming p = 0, meaning that the canopy fully covers the ground at the plant scale, Figure 7 shows two very different conditions in terms of LAI and C F , which could provide similar evaporation losses, since high C F (Figure 7a) could be counterbalanced by low LAI (Figure 7b), and vice versa. Δ$ * !9 Δ$ > Δ$ * Δ$ ℎ <!# (21) In both interception and evaporation models previously introduced, neither the canopy actually covering the field (cover fraction), nor the proportion of rain which falls through the canopy without striking a surface (throughfall), were taken into account. In the next section, the latter issues are considered and then the water mass balance is analyzed.

Water Mass Balance
The fraction of ground covered by the canopy [40] plays a fundamental role in estimating interception losses, as well as the proportion of rain that falls through the canopy without striking any surface. In the following, in order to analyze the water mass balance, CF denotes the fraction of ground covered by the canopy and p is the free throughfall coefficient [19,20]. Assuming p = 0, meaning that the canopy fully covers the ground at the plant scale, Figure 7 shows two very different conditions in terms of LAI and CF, which could provide similar evaporation losses, since high CF (Figure 7a) could be counterbalanced by low LAI (Figure 7b), and vice versa. By considering both CF and p, as weighting factors, for a gross rainfall depth R (mm), the water mass balance is described by the following relationship: where Rf (mm) is the portion of R that freely achieves the ground in between the plants, Rc (mm) is the fraction of R that achieves the canopy, and Rt (mm) is the throughfall. Thus, depending on the plant species, both p and CF affect Rc and could be simply determined by using RGB images [41,42]. According to Equation (22), to evaluate interception losses, in Equation (13) R has to be replaced by Rc.
By assuming CF = 1 and p = 0 (R = Rc), for dry initial condition, ICS0 = 0, for fixed parameters S = 0.8 mm, LAI = 4, m = 0.047, n = 0.657, Tex = 12 °C, and for three sequences of WS and DS, a simple application of the procedure is illustrated in Figure 8, where the denoted variables are also indicated. For simplicity, in the figure linear ICS and ∆E variations were assumed. Figure 8 shows that for the third DS a high air temperature (Tm = 20 °C) provides the condition ∆E > ICS, and only the available water stored on the canopy may evaporate (Δ$ * ). Moreover, for this step, according to the initial condition (t0, E0), the evaporation flux (i.e., the slope) is higher than those corresponding to the lower air temperatures. By considering both C F and p, as weighting factors, for a gross rainfall depth R (mm), the water mass balance is described by the following relationship: where R f (mm) is the portion of R that freely achieves the ground in between the plants, R c (mm) is the fraction of R that achieves the canopy, and R t (mm) is the throughfall. Thus, depending on the plant species, both p and C F affect R c and could be simply determined by using RGB images [41,42]. According to Equation (22), to evaluate interception losses, in Equation (13) R has to be replaced by R c .
By assuming C F = 1 and p = 0 (R = R c ), for dry initial condition, ICS 0 = 0, for fixed parameters S = 0.8 mm, LAI = 4, m = 0.047, n = 0.657, T ex = 12 • C, and for three sequences of WS and DS, a simple application of the procedure is illustrated in Figure 8, where the denoted variables are also indicated. For simplicity, in the figure linear ICS and ∆E variations were assumed. Figure 8 shows that for the third DS a high air temperature (T m = 20 • C) provides the condition ∆E > ICS, and only the available water stored on the canopy may evaporate (∆E * ). Moreover, for this step, according to the initial condition (t 0 , E 0 ), the evaporation flux (i.e., the slope) is higher than those corresponding to the lower air temperatures.
For fixed T m = 12 • C, the evaporation flux is greater for the second DS step that starts from a higher ICS 0 , which is near to the saturation compared to that of the first DS step. Similarly, for the second WS step, although the higher rainfall depth (R = 2 mm) with respect to the first WS step, ICS increases as in the first WS step, because of a higher initial condition ICS 0 , agreeing with the dynamic flux of the both considered interception/evaporation models. For fixed Tm = 12 °C, the evaporation flux is greater for the second DS step that starts from a higher ICS0, which is near to the saturation compared to that of the first DS step. Similarly, for the second WS step, although the higher rainfall depth (R = 2 mm) with respect to the first WS step, ICS increases as in the first WS step, because of a higher initial condition ICS0, agreeing with the dynamic flux of the both considered interception/evaporation models.
The flow chart displayed in Figure 9 describes the suggested procedure to calculate the interception/evaporation water losses, which will be used in the following application, where the condition to be imposed (Equation (21)) is also indicated. The water mass balance applied to the fraction of rainfall intercepted by the canopy, can be tested by checking the balance of the inflowing volumes as interception during the i WSs and the outflowing volumes, as evaporation from the canopy, during the j DSs, where of course the antecedent stored volumes ICS0 are involved: Once evaporation losses are calculated, the net rainfall Rn (mm) can be evaluated: The flow chart displayed in Figure 9 describes the suggested procedure to calculate the interception/evaporation water losses, which will be used in the following application, where the condition to be imposed (Equation (21)) is also indicated. For fixed Tm = 12 °C, the evaporation flux is greater for the second DS step that starts from a higher ICS0, which is near to the saturation compared to that of the first DS step. Similarly, for the second WS step, although the higher rainfall depth (R = 2 mm) with respect to the first WS step, ICS increases as in the first WS step, because of a higher initial condition ICS0, agreeing with the dynamic flux of the both considered interception/evaporation models.
The flow chart displayed in Figure 9 describes the suggested procedure to calculate the interception/evaporation water losses, which will be used in the following application, where the condition to be imposed (Equation (21)) is also indicated. The water mass balance applied to the fraction of rainfall intercepted by the canopy, can be tested by checking the balance of the inflowing volumes as interception during the i WSs and the outflowing volumes, as evaporation from the canopy, during the j DSs, where of course the antecedent stored volumes ICS0 are involved: Once evaporation losses are calculated, the net rainfall Rn (mm) can be evaluated: The water mass balance applied to the fraction of rainfall intercepted by the canopy, can be tested by checking the balance of the inflowing volumes as interception during the i WSs and the outflowing volumes, as evaporation from the canopy, during the j DSs, where of course the antecedent stored volumes ICS 0 are involved: Once evaporation losses are calculated, the net rainfall R n (mm) can be evaluated:

Interception Capacity LAI and Evaporation Measurements for Faba Bean
In order to show the results that can be obtained by the application of the simplified procedure, the 2019 rainfall ( Figure 2) and temperature data series of Fontanasalsa station were considered, and the required empirical parameters of Equation (19) were obtained by experimental measurements carried out for faba bean. Faba bean is a species of flowering plant in the pea and bean family Fabaceae, originated in the continent of Africa, which is widely cultivated as a crop for human consumption and also used as a cover crop.
For a single plant, planted in mid-November, and sampled after 95 days, the temporal variation of water loss by evaporation starting from the interception capacity, the number of leaves, # Leaves, and the corresponding surface area were measured. Saturation of the canopy was achieved by using sprinkler irrigation, for a fixed outdoor temperature, Starting from the interception capacity condition, after dripping has ceased, water loss by evaporation was measured by weighing during the time. The # Leaves (45), and the corresponding surface area (0.043 m 2 ), made it possible to calculate the cumulative evaporation volume, per unit leaf surface area, E (mm). Figure 10a plots the cumulate experimental E values during the time. This made it possible to calculate by a simple linear regression of the corresponding logarithmic values, the m and n parameters that are required to apply Equation (16). According to Babu et al. [36], Figure 10a shows that in a first stage evaporation rate is high, and then the evaporation rate decreases remaining almost constant in the time, for approximately t = 1 h (at least for this case), the behavior of which can be described by the power-law well. procedure, the 2019 rainfall ( Figure 2) and temperature data series of Fontanasalsa station were considered, and the required empirical parameters of Equation (19) were obtained by experimental measurements carried out for faba bean. Faba bean is a species of flowering plant in the pea and bean family Fabaceae, originated in the continent of Africa, which is widely cultivated as a crop for human consumption and also used as a cover crop.
For a single plant, planted in mid-November, and sampled after 95 days, the temporal variation of water loss by evaporation starting from the interception capacity, the number of leaves, # Leaves, and the corresponding surface area were measured. Saturation of the canopy was achieved by using sprinkler irrigation, for a fixed outdoor temperature, Tex = 12 °C.
Starting from the interception capacity condition, after dripping has ceased, water loss by evaporation was measured by weighing during the time. The # Leaves (45), and the corresponding surface area (0.043 m 2 ), made it possible to calculate the cumulative evaporation volume, per unit leaf surface area, E (mm). Figure 10a plots the cumulate experimental E values during the time. This made it possible to calculate by a simple linear regression of the corresponding logarithmic values, the m and n parameters that are required to apply Equation (16). According to Babu et al. [36], Figure 10a shows that in a first stage evaporation rate is high, and then the evaporation rate decreases remaining almost constant in the time, for approximately t = 1 h (at least for this case), the behavior of which can be described by the power-law well.
Of course, it should be noted that there are other secondary factors not considered in the present approach, such as the plant structure, the leaf angle and the light interception, although not easy to consider, which may affect the considered parameters [43], especially when upscaling is needed. Figure 10. For Field bean, (a) relationship between experimental evaporation losses per unit LAI, E/LAI, during the time t (h), starting from the interception capacity, S (mm), and (b) assumed temporal variation of LAI and S (mm), from the day of planting (mid-November), considered for the application (Figure 11).
For the purpose of the present investigation, the relationship between the interception capacity and LAI was considered in order to extend to 2009, the corresponding relationships displayed in Figure 10a, which are related to the limited experimental LAI value. To consider the LAI temporal variations, some data for faba bean available in the literature were considered. The figure also illustrate the water mass balance (red line) that was checked by using Equation (23), and the gross and net rainfall depth, R and Rn (Equation (24)), respectively (secondary axis).
For faba bean planted in mid-November, as in this study, in El-Bosaily farm in the Northern coast of Egypt, Hegab et al. [44] measured a maximum LAI value equal to 5.5. This value was assumed for the mid-season stage according to FAO56 [35], which provides general lengths for four distinct growth stages, initial (ini.), crop development (crop dev.), mid-season (mid. s.) and late season (late), for various, crops, types of climates and locations.
Of course, it should be noted that there are other secondary factors not considered in the present approach, such as the plant structure, the leaf angle and the light interception, although not easy to consider, which may affect the considered parameters [43], especially when upscaling is needed.
For the purpose of the present investigation, the relationship between the interception capacity and LAI was considered in order to extend to 2009, the corresponding relationships displayed in Figure 10a, which are related to the limited experimental LAI value. To consider the LAI temporal variations, some data for faba bean available in the literature were considered.
For faba bean planted in mid-November, as in this study, in El-Bosaily farm in the Northern coast of Egypt, Hegab et al. [44] measured a maximum LAI value equal to 5.5. This value was assumed for the mid-season stage according to FAO56 [35], which provides general lengths for four distinct growth stages, initial (ini.), crop development (crop dev.), mid-season (mid. s.) and late season (late), for various, crops, types of climates and locations.
For faba bean, FAO56 [35] reports ini. = 90, crop dev. = 45, mid.s. = 40, late s. = 0 days, thus no late season was considered since it was assumed that after the mid-season the faba bean was fully removed from the field. For the four growth stages, Figure 10b shows the LAI variations versus the days from planting that made it possible the applications described in the next section to be performed. In Figure 10b can be observed that it was assumed that the crop development stage, after 90 days from planting, starts with LAI = 0.5.
According to Brisson et al. [45], the interception capacity was assumed S = 0.2 LAI. Thus, with increasing LAI during crop growth and decreasing LAI during senescence, S will increase and decrease, respectively. More recently, this relationship between S and LAI was also considered by Kozak et al. [46]. Figure 10b also graphs the S variation during the days from planting. The interception capacity in the day of sampling (cross symbol), i.e., after 95 days from the day of planting, was equal to 0.202, not too far from that experimentally measured (0.156).
It should be noted that other S vs. LAI relationships could be used in the presented methodology, and that specific methods, as that proposed by Aston [47] who also found a linear S vs. LAI relation, could be applied in order to experimentally determine such relationships that depend on the considered crops.

Example of Application
For the considered crop described in the previous section (Figure 10a,b), for C F = 1 and p = 0, and for the 2019 rainfall data series, interception and evaporation losses were calculated according to the described procedure. Figure 11 plots the detailed results of the involved cumulative water balance components, i.e., the rainfall depth, R (mm), the interception depth at the end of each wet spell, ICS, and the evaporation losses, ∆E * . The latter made it possible to evaluate the antecedent stored volume before each WS, ICS 0 = ICS − ∆E * .
The figure also illustrate the volume water mass balance (red line) that was checked by using Equation (23). Of course, the interception process mostly occurs during the growing season when it rains. For the considered year, 2009, the evaporation loss by canopy achieved 37.6 mm. Figure 11 also plots the net rainfall (dashed line), calculated according to Equation (24).
It is interesting to observe that the rainiest periods (1-110 and 250-365 Julian day) are not associated with the periods with the highest evaporation losses (1-110 Julian day). Of course, this is explained by the rainfall distribution, which has to be analyzed with respect to the growing period of the considered crop, playing a fundamental role in detecting evaporation losses by the rainfall intercepted by the canopy. Evaporation losses for the considered cover crop are 4.5% with respect to the yearly rainfall, and are thus lower than the minimum found in other studies [28], despite different cover crops being considered, whereas interception losses are of course higher and equal to 10.58%.
However, this author is of the opinion that comparison of these values with those obtained in other studies is not very meaningful, due to the high interactions between climate variability, rainfall distribution, and cover crop management (growing season, date of planting, etc.) that should be analyzed to explain the impact of inter-annual variability of interception/evaporation losses [48]. In this sense, the procedure described in this work could be useful to study in deep these interactions, also accounting for the other parameters as the cover fraction, the date of planting [42], the throughfall index and different rainfall regimes and their changes [33,49] that were not considered in this work.

Conclusions
In order to derive the evaporation losses by wet canopy, the suggested procedure combines a modified interception model proposed by Merrian, which is applied during rainfall wet spells, and a simple power-law equation to model evaporation during the dry spells. This simple approach makes it possible to estimate the evaporation losses during continuous simulations, and requires few parameters that consolidate climate and crop conditions. Moreover, it is shown that the Merrian model can be derived by considering a simple linear storage model that makes it possible to account of the antecedent intercepted volume, which is useful for applications. The crop considered in the application is the faba bean that was described according to the general lengths of four distinct growth stages considered in FAO56, whereas LAI and interception capacity were obtained from literature. Since the required parameters are few, this simple approach could be applied when a rough estimate of evaporation loss by wet canopy is necessary, in the absence of a detailed characterization of canopy and climate. Due to simplified schemes, the procedure can be easily applied at large scale, in order to study the important role of rainfall regime and crop growing stages, on this important component of the hydrologic cycle, and it could be suitable to be implemented according to a probabilistic line of approach.
Funding: This research received no external funding. Data Availability Statement: All data, models, and code, generated or used during the study appear in the published article.